Initial data for binary black holes: the conformal thin-

sandwich puncture method

Mark D. Hannam

UTB Relativity Group SeminarSeptember 26, 2003

Overview: the smallest picture possible

• We want to simulate a (realistic) binary black hole collision. To do that,

1. Rewrite Einstein’s equations as a Cauchy problem2. Set up initial data for two black holes in orbit3. Evolve the system.

• Problems: we can’t do (2) or (3) very well.• Partial solution: try to create good initial data

close to the interesting physics…• Describe two black holes in quasi-circular, quasi-

equilibrium orbit just before they plunge together.

)(2

1ijjiijtij N

K

))((222 dtdxdtdxdtNds jjiiij

Initial data: ijij K,

3,2,1,0,,82

1 TRgR Space and time are mixed…

ijjiijijt

ijij

lijl

ljilijl

l

jiljilijijijt

ililil

ijij

NK

SSGN

KKK

NKKKKRNK

jGKK

GKKKR

2

))(2(4

)2(

8)(

162 Initial value constraints

Evolutionequations

ij'

ijt

tt

tN

ti

What quantities are constrained?

12 independent components - 4 constraint equations

8 free quantities: 4 dynamical 4 gauge

Which are which?

Use a conformal decomposition…

ijij K,

Conformal thin-sandwich decomposition

NNiiijij

~~~ 64

KKAAKAK ijijijijij

~~

3

1 2

ijtijk

kijijjiij

ijijij

uL

uLN

A

~~,~~

3

2~~~

,~~~

2

1~

KKKSGAANN

jNuN

NKNNL

GAAKR

tllij

ij

iijj

ij

ijiL

ijij

~

12

5)2(2

~~

8

7)

~()

~(

~

~16~

~1~~~~

3

4~ln

~~~

2~~~~

52548772

106

578

1258

1

8

12

From ... Kt

CTS: the essentials

• Free data:

• Solve for:

• Construct:

K

K

tijt

ij

~

~

Ni ~,,

ijij K,

“Easy” examples• Schwarzschild (single stationary black hole):

• Brill-Lindquist (multiple stationary black holes)

ed...)undetermin are (,~21

~

0~

0,~21

0,0,0~,~

7i

N

i i

i

iji

N

i i

i

tijtijij

cr

cN

Ar

m

KKf

0~

0,~21

~,~2

1

0,0,0~,~

7

iji

tijtijij

Ar

MN

r

M

KKf

Orbits in the CTS decomposition

• In a corotating reference frame, the black holes will be almost stationary.

• Choose

• These choices are physically motivated– Free data choices in old decompositions were

made for convenience

0,0~ Ku tij

CTS solutions

• Gourgoulhon, Grandclément, and Bonazzola (GGB), 2001.– Solved with– Excised regions containing singularities– Employed boundary conditions on excised

surfaces(…there were inconsistencies here)

• I want to avoid inner boundary conditions

Puncture method.

0,~ Kfijij

CTS-puncture approachRecall Brill-Lindquist solution:

Extend to .

The shift has no singular part– What corresponds to black holes with Pi and Si ?

vr

cN

N

i i

i ~21

~ 7

ur

mN

i i

i ~21

7~N

(what are ci?)

KKAAvr

cv l

lijij

N

i i

i

~

12

5~~

8

7~2

1~ 52582

Hamiltonian constraint:

Constant-K equation:

Solve for v

2572

12

1~~

8

1~KAAu ij

ij

Regular if

3

6

~~

~~~~

rK

rAA ijij

Solve for u

Issues: Slicing choices(for one black hole)

• Two principle choices:

• Schwarzschild

– But: on some surface…

• Estabrook (N = 1).

but

– This is a “dynamical” slicing!– The stationary Schwarzschild black hole will APPEAR

to have dynamics

• This isn’t necessarily fatal to the method

[Ref: MDH, C.R. Evans, G.B. Cook, T.W. Baumgarte, gr/qc-0306028]

mc

mc

0~ N ijij L

NA ~

~2

1~

mc 1

~ N 0~ ijt A

r

cN ~2

1~ 7

r

mN ~2

1~ 7

r

mN ~2

1~ 7

Issue #2: The shift vectorConditions at the puncture?

• The analytic, singular part of the conformal factor gave us a black hole solution, without the need for inner boundary conditions

• There is no known analytic part of the shift for a black hole with non-zero Pi and Si, and the puncture form of the lapse.

• We need to impose suitable conditions at the puncture.

• Methods to date do not give convergent results…

Future work

• Convert code to Cactus, where much greater resolution is possible – Maybe the momentum constraint solver will

converge.

• Construct data with an everywhere positive lapse

• Examine the level of stationarity of quasi-circular orbits (located by, for example, the effective potential method)– Maybe the “Estabrook” lapse choice is Ok.